To study the steady-state properties of the Floquet three-level system, only the data at the end of each modulation period (i.e., the time evolution step is \(\tau \) ) is observed. The dynamics start from the ground state \(|0\rangle \) , we show the contour map of \(\protect \rho _{11}\)  as a function of the probe field detuning \(\Delta \)  and the modulation period \(\protect \tau \)  in Fig. REF (b). One finds that each peak in Fig.REF (b) splits into two when an added control field couples \(|1\rangle \)  to \(|2\rangle \) , resulting in multiple transparency windows, which is similar to the multi-chromatic ATS with multi-tone control fields studies in the un-modulated systems [1]}, [2]}, [3]}. The positions of the peaks, as the white dashed lines shown in Fig. REF (b), could be demonstrated to be exactly the maximal constructive interference of transitions \(|+\rangle \leftrightarrow |0\rangle \)  and \(|-\rangle \leftrightarrow |0\rangle \) , which could be achieved by calculating relative phases \(\phi _{0\pm }\)  in one modulation period. Here \(|\pm \rangle =(|2\rangle \pm |1\rangle )/\sqrt{2}\)  is the dressed levels induced by the strong control field. During the controlled half-period, the system is in the dressed basis \(|\pm \rangle \)  with eigenenergy \((\Delta \pm \Omega _{c})/2\) . The relative phases accumulated in
the first half period are \(\phi _{0\pm }^{\prime }=(\Delta \pm \Omega _{c}/2)\tau /2\) . During the non-controlled half period,
the system is in bare basis. Driven by a probe field with detuning \(\Delta \) , the relative phases \(\phi _{0\pm }^{\prime \prime }\)  accumulated are \(\phi _{0+}^{\prime \prime }=\phi _{0-}^{\prime \prime }\approx \Delta \tau /2,\) 
where \(\Omega _{p}\ll \Delta \)  is assumed. Then the total relative
phases in one period are \(\phi _{0\pm }=\phi _{0\pm }^{\prime }+\phi _{0\pm }^{\prime \prime }=(\Delta \pm \Omega _{c}/4)\tau \) . The maximal
interference requires \(\phi _{0\pm }=2n\pi \) , \(~(n=0,1,2\cdots )\) .
Therefore, the constraints for absorption peaks are several curves \(\Delta =2n\pi \omega \pm \Omega _{c}/4\) , as the white dashed lines shown in Fig. REF (b). The multiple transparency windows may have promising applications in quantum memory [4]}, ground-state cooling [5]} and so on.
